Tüm alıştırma soruları

Complete the passage by entering the punctuation mark that correctly sets off the nonessential element in the sentence.

Aşağıdaki boşlukları doldurun

In 1928, Scottish biologist Alexander Fleming discovered penicillin, the world's first effective antibiotic, quite by accident. While studying *Staphylococcus* bacteria in his London laboratory, Fleming noticed that a contaminated Petri dish—which had been left unattended near an open window had grown a mold that successfully inhibited the growth of the surrounding bacteria.

Cevabı ve açıklamayı göster

Cevap

An em dash (—) is required to close the nonessential relative clause and match the opening em dash.

The sentence features a nonessential relative clause ('which had been left unattended near an open window') that interrupts the flow between the subject ('Petri dish') and its verb phrase ('had grown'). Standard English requires nonessential elements to be set off with matching punctuation. Because the clause is introduced with an em dash, a matching em dash must be used to close it.

Adım Adım Çözüm

1

Identify the nonessential element in the sentence.

The clause starting with 'which' and ending at the blank is nonessential to the main clause of the sentence.

Removing the clause 'which had been left unattended near an open window' does not change the core grammatical structure or primary meaning of the sentence.

2

Identify the punctuation mark that introduces the nonessential element.

An em dash (—) is used immediately before the relative pronoun 'which'.

Standard punctuation conventions require nonessential elements to be set off with matching pairs of punctuation marks.

3

Determine the correct punctuation to end the nonessential element.

An em dash (—) must be placed in the blank.

Since the nonessential clause begins with an em dash, it must also end with an em dash to maintain grammatical symmetry.

Anahtar Kavram

Punctuation of Nonessential Elements

Tahmini Süre:1m 0s

Soru 1182Soru →

Which word completes the passage so that it conforms to the conventions of Standard English?

Aşağıdaki boşlukları doldurun

In their recent excavation of a medieval settlement, a team of Swedish archaeologists discovered a remarkably well-preserved wooden ship. To accurately estimate the age of the wooden artifact, the research team planned to conduct radiocarbon dating on the organic fibers, analyze the tree ring patterns of the timber, and the historical records of local maritime trade from that era.

Cevabı ve açıklamayı göster

Cevap

A base verb (such as 'consult' or 'examine') that is parallel to the other base verbs in the series, 'conduct' and 'analyze'.

To maintain parallel structure in the list of actions introduced by 'planned to' (conduct..., analyze..., and [blank]), the blank must contain a base verb (such as 'consult' or 'examine'). This matches the base verb forms of the preceding coordinates 'conduct' and 'analyze'.

Adım Adım Çözüm

1

Identify the list of coordinate elements in the sentence.

The sentence lists three actions that the research team planned to perform: (1) 'conduct radiocarbon dating...', (2) 'analyze the tree ring patterns...', and (3) '{{blank_1}} the historical records...'.

Recognizing the structure of the list is essential to verifying grammatical consistency.

2

Analyze the grammatical form of the completed items in the list.

The first two items in the list begin with the base verbs 'conduct' and 'analyze', following the initial helper verb phrase 'planned to'.

The grammatical form of the initial items dictates the form required for the final item.

3

Determine the grammatical form required for the blank to maintain parallel structure.

The blank must be completed with a base verb (e.g., 'consult') to remain parallel with 'conduct' and 'analyze'. Present participles or verbs preceded by 'to' would disrupt this parallel structure.

Standard English conventions require all coordinate elements in a series to share the same grammatical form.

Anahtar Kavram

Parallel structure in a list of verbs requires all verbs to share the same grammatical form. When a helper verb like 'to' precedes the first item and is not repeated, the subsequent items must be base verbs.

Tahmini Süre:1m 0s

Soru 1183Soru →

For all $x \ge 0$ , the expression $(3x^{\frac{1}{2}} + 2)(2x^{\frac{1}{2}} - 5)$ is equivalent to $ax - b\sqrt{x} - 10$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

Cevabı ve açıklamayı göster

Cevap: 17

Cevap

17

Expanding the expression

(3x^{\frac{1}{2}} + 2)(2x^{\frac{1}{2}} - 5)

yields

6x - 15x^{\frac{1}{2}} + 4x^{\frac{1}{2}} - 10

. Combining like terms gives

6x - 11\sqrt{x} - 10

. Comparing this result to the given equivalent form

ax - b\sqrt{x} - 10

shows that

a = 6

and

b = 11

. Therefore, the value of

a + b

is

6 + 11 = 17

.

Adım Adım Çözüm

1

Multiply the binomials

(3x^{\frac{1}{2}} + 2)(2x^{\frac{1}{2}} - 5)

using FOIL.

(3x^{\frac{1}{2}})(2x^{\frac{1}{2}}) - 5(3x^{\frac{1}{2}}) + 2(2x^{\frac{1}{2}}) - 10

Expanding the product allows us to write it in the standard simplified form.

2

Simplify the products and combine the like terms.

6x - 15x^{\frac{1}{2}} + 4x^{\frac{1}{2}} - 10 = 6x - 11x^{\frac{1}{2}} - 10

This simplifies the expression so we can compare it directly to the target form.

3

Rewrite

x^{\frac{1}{2}}

as

\sqrt{x}

and compare the expression to

ax - b\sqrt{x} - 10

.

6x - 11\sqrt{x} - 10 = ax - b\sqrt{x} - 10

, which gives

a = 6

and

b = 11

.

Matching corresponding coefficients allows us to determine the values of the constants

a

and

b

.

4

Calculate the sum of

a

and

b

.

a + b = 6 + 11 = 17

The question requires finding the value of

a + b

.

Anahtar Kavram

To determine equivalence between algebraic expressions, expand the terms using the distributive property, simplify, and equate the corresponding coefficients of the like terms.

Soru 1184Soru →

A local logistics company charges a flat account setup fee of $5$ dollars plus $4$ dollars per pound for the first $6$ pounds of a package shipment. For any weight beyond $6$ pounds, the company charges $3$ dollars per pound. If the total cost to ship a package is $50$ dollars, what is the total weight of the package, in pounds?

7

9

13

1

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

The correct answer is 13, which represents the total weight of the package in pounds.

To find the total weight, we first determine the cost of the first

6

pounds, which is the

5

dollars flat fee plus

4

dollars per pound, totaling

29

dollars. Subtracting this baseline cost from the total cost of

50

dollars leaves

21

dollars for the weight exceeding

6

pounds. Since the rate for the excess weight is

3

dollars per pound, the excess weight is

21

divided by

3

, which is

7

pounds. Adding the initial

6

pounds gives a total weight of

13

pounds.

Adım Adım Çözüm

1

Calculate the total cost of shipping the first 6 pounds of the package.

The cost for the first

6

pounds, including the flat setup fee, is

5 + 6 \times 4 = 29

dollars.

This establishes the baseline cost before the rate of

3

dollars per pound is applied to any weight exceeding

6

pounds.

2

Set up a linear equation representing the total shipping cost of 50 dollars for a package of weight w, where w is greater than 6.

The equation is

29 + 3(w - 6) = 50

.

The total cost is the sum of the cost for the first

6

pounds (

29

dollars) and the cost of the excess weight (

w - 6

) at the rate of

3

dollars per pound.

3

Solve the linear equation for w by isolating the variable.

3(w - 6) = 21 \implies w - 6 = 7 \implies w = 13

.

Subtracting

29

from both sides and then dividing by

3

isolates the term representing the excess weight, and adding

6

yields the total weight.

Anahtar Kavram

Linear Equations in One Variable

Soru 1185Soru →

Which word completes the passage so that it conforms to the conventions of Standard English?

Aşağıdaki boşlukları doldurun

In his study of ancient agricultural practices, archaeologist Liam O'Connor argues that the transition to farming in the Fertile Crescent was motivated not by a sudden climatic shift but a gradual decline in wild food resources. According to O'Connor, early hunter-gatherers were forced to cultivate crops as their traditional foraging grounds became depleted, demonstrating that necessity, rather than choice, drove agricultural innovation.

Cevabı ve açıklamayı göster

Cevap

by

The correct answer is 'by'. The sentence uses the contrastive construction 'not [X] but [Y]' to contrast two potential causes of the agricultural transition. To maintain parallel structure, the grammatical form of both parts of the contrast must be identical. Because the first part of the contrast is a prepositional phrase introduced by the preposition 'by' ('not by a sudden climatic shift'), the second part must also be a prepositional phrase introduced by the same preposition ('but by a gradual decline').

Adım Adım Çözüm

1

Identify the parallel elements and grammatical structures in the sentence.

The sentence presents a contrast using the structure 'not [element 1] but [element 2]'. The first element is 'by a sudden climatic shift'.

Recognizing the parallel construction helps determine the grammatical pattern required for the second element.

2

Analyze the grammatical structure of the first element.

The phrase 'by a sudden climatic shift' is a prepositional phrase headed by the preposition 'by'.

Parallel structure requires that elements joined by coordinating or correlative conjunctions have the same grammatical form.

3

Complete the second element to match the first.

The second element must also be a prepositional phrase headed by the preposition 'by', resulting in 'by a gradual decline'.

Using the preposition 'by' after 'but' ensures that both elements are grammatically parallel.

Anahtar Kavram

Parallel Structure with Prepositional Phrases

Soru 1186Soru →

In the $xy$ -plane, the graph of the linear equation $ax + by = c$ , where $a$ , $b$ , and $c$ are non-zero constants, has a positive slope and a negative $y$ -intercept. If the line passes through the point $(b, a)$ , which of the following inequalities must be true?

ab > 0

ac > 0

bc > 0

ac < 0

Cevabı ve açıklamayı göster

Cevap: $ac > 0$

Cevap

The inequality

ac > 0

must be true.

The correct inequality is

ac > 0

. Rearranging

ax + by = c

into slope-intercept form gives

y = -\frac{a}{b}x + \frac{c}{b}

. A positive slope means

-\frac{a}{b} > 0

, so

\frac{a}{b} < 0

, indicating that

a

and

b

have opposite signs (

ab < 0

). A negative

y

-intercept means

\frac{c}{b} < 0

, indicating that

b

and

c

have opposite signs (

bc < 0

). Substituting the point

(b, a)

into the line's equation gives

2ab = c

. Since

ab < 0

, it follows that

c < 0

. Using the relation

bc < 0

with

c < 0

, we find

b > 0

. Using

ab < 0

with

b > 0

, we find

a

is negative. Since both

a

and

c

are negative, their product

ac

must be positive.

Adım Adım Çözüm

1

Rearrange the standard form equation

ax + by = c

into slope-intercept form.

y = -\frac{a}{b}x + \frac{c}{b}

To identify the expressions for the slope and the

y

-intercept in terms of the constants

a

,

b

, and

c

.

2

Determine the sign relationships between the coefficients based on the given signs of the slope and

y

-intercept.

Since the slope is positive,

-\frac{a}{b} > 0 \implies ab < 0

. Since the

y

-intercept is negative,

\frac{c}{b} < 0 \implies bc < 0

.

This establishes that

a

and

b

have opposite signs, and

b

and

c

have opposite signs.

3

Substitute the coordinates of the point

(b, a)

into the original equation

ax + by = c

.

a(b) + b(a) = c \implies 2ab = c

Since the point lies on the line, substituting its coordinates must satisfy the equation.

4

Analyze the sign of

c

using the relation

c = 2ab

.

Since

ab < 0

, it follows that

2ab < 0

, which means

c < 0

.

To determine whether the constant

c

is positive or negative.

5

Determine the individual signs of

b

and

a

.

Since

bc < 0

and

c < 0

,

b

must be positive (

b > 0

). Since

ab < 0

and

b > 0

,

a

must be negative (

a < 0

).

To find the sign of each coefficient individually.

6

Determine the sign of the product

ac

.

Since both

a < 0

and

c < 0

, their product

ac

must be positive (

ac > 0

).

To evaluate which inequality must be true.

Anahtar Kavram

Analyzing slope, intercepts, and points on a linear equation in two variables using sign analysis of its constants.

Alternatif Yöntem

Instead of solving for the individual signs of all variables, we can choose test values that satisfy the conditions. For example, let

a = -1

and

b = 2

, which gives a positive slope of

-\frac{a}{b} = \frac{1}{2}

. Since the line passes through

(b, a) = (2, -1)

, we can find

c

:

ax + by = c \implies (-1)(2) + (2)(-1) = -4

, so

c = -4

. The

y

-intercept is

\frac{c}{b} = \frac{-4}{2} = -2

, which is negative, satisfying all conditions. Evaluating the product of

a

and

c

gives

(-1)(-4) = 4 > 0

, confirming that the product of these constants must be positive.

Tahmini Süre:3m 0s

Soru 1187Soru →

In the $xy$ -plane, a line with a positive slope $m$ passes through the point $(4, -3)$ and intersects the $x$ -axis at $(p, 0)$ and the $y$ -axis at $(0, q)$ , where $p$ and $q$ are non-zero constants. If $p + q = 5$ , what is the value of $m$ ?

Cevabı ve açıklamayı göster

Cevap: 0.5

Cevap

The value of

m

is

0.5

(or

1/2

).

To find the slope

m

, write the equation of the line in point-slope form:

y - (-3) = m(x - 4)

, which simplifies to

y = mx - 4m - 3

. The

y

-intercept is found by setting

x = 0

, giving

q = -4m - 3

. The

x

-intercept is found by setting

y = 0

, giving

p = 4 + \frac{3}{m}

. Substituting these expressions into

p + q = 5

gives

4 + \frac{3}{m} - 4m - 3 = 5

, which simplifies to

-4m - 4 + \frac{3}{m} = 0

. Multiplying this equation by

-m

yields the quadratic equation

4m^2 + 4m - 3 = 0

. Factoring the quadratic gives

(2m - 1)(2m + 3) = 0

. Since the line is defined to have a positive slope, the value of

m

must be positive, which is

0.5

(or

1/2

).

Adım Adım Çözüm

1

Write the general equation of a line passing through

(4, -3)

with slope

m

.

y = mx - 4m - 3

Using the point-slope form

y - y_1 = m(x - x_1)

with

(x_1, y_1) = (4, -3)

establishes the relationship between

x

,

y

, and

m

.

2

Determine the value of

q

by finding the

y

-intercept of the line.

q = -4m - 3

The

y

-intercept

(0, q)

occurs when

x = 0

.

3

Determine the value of

p

by finding the

x

-intercept of the line.

p = 4 + \frac{3}{m}

The

x

-intercept

(p, 0)

occurs when

y = 0

.

4

Substitute the expressions for

p

and

q

into the given equation

p + q = 5

.

4 + \frac{3}{m} - 4m - 3 = 5

This sets up a single equation with the variable

m

using the given relationship.

5

Simplify the equation and rewrite it in standard quadratic form.

4m^2 + 4m - 3 = 0

Multiplying the simplified expression

-4m - 4 + \frac{3}{m} = 0

by

-m

eliminates the fraction and aligns it to standard quadratic form.

6

Factor the quadratic equation to find the possible values of

m

.

m = 0.5

or

m = -1.5

Factoring

(2m - 1)(2m + 3) = 0

gives the roots of the equation.

7

Filter the results based on the positive slope constraint.

m = 0.5

The problem states that the line has a positive slope, so we must reject the negative root.

Anahtar Kavram

Using linear intercepts to solve system constraints

Alternatif Yöntem

Instead of using point-slope form, use the intercept form of a linear equation:

\frac{x}{p} + \frac{y}{q} = 1

. Since the line passes through the point

(4, -3)

, substitute

x = 4

and

y = -3

to get

\frac{4}{p} - \frac{3}{q} = 1

. Given

p + q = 5

, substitute

p = 5 - q

into the equation to get

\frac{4}{5 - q} - \frac{3}{q} = 1

. Multiply both sides by the common denominator

q(5 - q)

to obtain

4q - 3(5 - q) = q(5 - q) \implies 7q - 15 = 5q - q^2 \implies q^2 + 2q - 15 = 0

. Factoring gives

(q + 5)(q - 3) = 0

. This yields

q = -5

or

q = 3

. If

q = 3

, then

p = 2

, and the slope is

m = -\frac{q}{p} = -\frac{3}{2}

(negative). If

q = -5

, then

p = 10

, and the slope is

m = -\frac{q}{p} = -\frac{-5}{10} = 0.5

(positive). Since the slope must be positive, the slope of the line is

0.5

.

Tahmini Süre:2m 30s

Soru 1188Soru →

In a certain video game, a player earns 15 points for completing a level and loses 3 points for each hint they use. If a player wants to score at least 6 points on a level, what is the maximum number of hints they can use?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The maximum number of hints the player can use is 3.

To find the maximum number of hints, we construct the inequality representing the player's score:

15 - 3h \geq 6

, where

h

is the number of hints used. Subtracting 15 from both sides gives

-3h \geq -9

. Dividing both sides by

-3

and reversing the inequality sign yields

h \leq 3

. This means the player can use at most 3 hints to achieve a score of at least 6 points. Thus, the maximum number of hints is 3.

Adım Adım Çözüm

1

Set up the inequality representing the score requirement.

15 - 3h \geq 6

The starting score is 15, and 3 points are lost for each hint

h

. The final score must be at least (greater than or equal to) 6.

2

Subtract 15 from both sides of the inequality.

-3h \geq -9

This isolates the variable term on the left side of the inequality.

3

Divide both sides by -3 and flip the inequality sign.

h \leq 3

Dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.

Anahtar Kavram

Solving multi-step linear inequalities with negative coefficients

Tahmini Süre:45s

Soru 1189Soru →

Astrochemists studying the interstellar medium have detected various complex organic molecules, including formaldehyde and methanol, inside massive star-forming regions. These dense molecular clouds contain high concentrations of cosmic dust and ______ shield the delicate compounds from the destructive ultraviolet radiation emitted by nearby stars, thereby allowing the molecules to persist over long periods.

Which choice completes the text so that it conforms to the conventions of Standard English?

gas, they

gas they

gas; they

gas, but they

Cevabı ve açıklamayı göster

Cevap: gas; they

Cevap

gas; they

The option containing the semicolon correctly links two independent clauses. The clause before the blank ('These dense molecular clouds contain high concentrations of cosmic dust and gas') and the clause after the blank ('they shield the delicate compounds...') are both complete thoughts that can stand alone as sentences. A semicolon is a standard and grammatically correct way to join two independent clauses without a coordinating conjunction.

Adım Adım Çözüm

1

Identify the structure of the clauses around the blank.

The text before the blank ('These dense molecular clouds contain high concentrations of cosmic dust and gas') and the text after the blank ('they shield the delicate compounds...') are both independent clauses.

Understanding clause boundaries helps determine the appropriate punctuation or conjunction needed to link them.

2

Evaluate how two independent clauses can be grammatically linked.

Two independent clauses can be joined by a period, a semicolon, or a comma followed by a coordinating conjunction (such as 'and', 'but', or 'so').

This establishes the rules of Standard English conventions for clause linking.

3

Test the available options to see which one correctly and logically links the clauses.

The option containing a semicolon correctly links the two clauses. The option with a comma alone creates a comma splice, the option with no punctuation creates a run-on, and the option with 'but' introduces an illogical contrast.

Selecting the option that uses correct punctuation and maintains logical coherence ensures grammatical correctness.

Anahtar Kavram

Two independent clauses must be separated by a period, joined by a semicolon, or linked with a comma and a coordinating conjunction.

Soru 1190Soru →

Although marine biologist Rachel Carson is best known today for her 1962 book *Silent Spring*, her earlier trilogy of books about the ocean—which detailed marine ecosystems with poetic precision—______ her reputation as a pioneering science writer long before that later landmark work sparked the modern environmental movement. Indeed, the commercial success of these early works allowed Carson to resign from her government job and focus entirely on her writing.

Which choice completes the text so that it conforms to the conventions of Standard English?

establishes

had established

has established

will establish

Cevabı ve açıklamayı göster

Cevap: had established

Cevap

The correct answer is the past perfect verb form 'had established'.

The past perfect verb form 'had established' correctly describes an action that was completed in the past before another past action (indicated by the simple past verb 'sparked').

Adım Adım Çözüm

1

Identify the temporal framework of the passage by looking at neighboring verbs like 'sparked' and 'allowed', which establish a past-time context.

The passage describes historical events that occurred in the past.

This helps determine the base time period of the passage.

2

Analyze the sequence of events. The establishing of Carson's reputation occurred before the sparking of the environmental movement in the past.

The action in the blank happened prior to the action of 'sparked'.

This determines the aspect/tense relation of the verb.

3

Select the past perfect tense ('had established') to correctly show that one past action was completed prior to another past action.

The past perfect tense 'had established' is selected.

This conforms to Standard English conventions for sequencing past events.

Anahtar Kavram

Verb Tense, Aspect, and Mood

Tahmini Süre:1m 0s

Soru 1191Soru →

Many carnivorous plants rely on passive traps to capture their food; however, the Venus flytrap (*Dionaea muscipula*) employs an active mechanism to ensure it does not waste energy on false alarms. When an insect touches the trap's sensory hairs, the plant does not close immediately on the _______ must detect a second touch within twenty seconds to trigger closure.

Which choice completes the text so that it conforms to the conventions of Standard English?

prey; it

prey, it

prey it

prey; because it

Cevabı ve açıklamayı göster

Cevap: prey; it

Cevap

prey; it

The correct option correctly uses a semicolon to separate two independent clauses. The clause before the blank is independent, and the pronoun 'it' starts a second independent clause, making the semicolon the appropriate punctuation mark.

Adım Adım Çözüm

1

Analyze the clauses surrounding the blank space.

The clause before the blank ('When an insect touches the trap's sensory hairs, the plant does not close immediately on the prey') is an independent clause because it contains a subject and a verb and can stand alone as a complete sentence. The clause after the blank ('it must detect a second touch within twenty seconds to trigger closure') is also an independent clause.

Identifying clause types determines the correct punctuation needed to connect them.

2

Determine the appropriate punctuation to link two independent clauses.

Two independent clauses can be joined by a semicolon (without a coordinating conjunction) or a period, or by a comma followed by a coordinating conjunction (like 'and' or 'but'). The option with a semicolon followed by a pronoun ('prey; it') is grammatically correct.

Standard English conventions require a semicolon or a coordinating conjunction to link independent clauses.

Anahtar Kavram

Using semicolons to connect independent clauses

Tahmini Süre:45s

Soru 1192Soru →

A local store sells two types of coffee beans: Arabica and Robusta. On Monday, the store sold $8$ pounds of Arabica coffee and $5$ pounds of Robusta coffee for a total of $\$62.00$ . On Tuesday, the store sold $4$ pounds of Arabica coffee and $7$ pounds of Robusta coffee for a total of $\$58.00$ . What is the cost, in dollars, of one pound of Robusta coffee?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The cost of one pound of Robusta coffee is 6 dollars.

The correct answer is 6. By setting up the system of equations

8a + 5r = 62

and

4a + 7r = 58

, we can multiply the second equation by 2 to get

8a + 14r = 116

. Subtracting the first equation from this yields

9r = 54

, which simplifies to

r = 6

.

Adım Adım Çözüm

1

Define variables for the price per pound of Arabica coffee (

a

) and Robusta coffee (

r

), then set up a system of linear equations based on the given store sales information.

Equation 1:

8a + 5r = 62

Equation 2:

4a + 7r = 58

This translates the word problem context into a solvable system of mathematical equations.

2

Multiply the second equation by

2

to make the coefficients of

a

equal in both equations.

8a + 14r = 116

This prepares the system for solving by the elimination method.

3

Subtract the first equation (

8a + 5r = 62

) from the new equation (

8a + 14r = 116

) to eliminate

a

and solve for

r

.

9r = 54

, which simplifies to

r = 6

Subtracting the equations eliminates one variable, leaving a single-variable equation that is easily solved.

Anahtar Kavram

Solving systems of two linear equations in two variables using the elimination method.

Tahmini Süre:1m 30s

Soru 1193Soru →

In the system of equations below, $c$ is a constant.

\begin{aligned} 3x - 2y &= 3c - 1 \\ x + 3y &= 2c - 8 \end{aligned}

If the solution $(x, y)$ to the system satisfies $x + y = 2$ , what is the value of $c$ ?

4

16

3

-1

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The correct value of the constant

c

is 4.

To find the value of

c

, the linear constraint

x + y = 2

is rearranged to

x = 2 - y

. Substituting this expression into the first equation yields

3(2 - y) - 2y = 3c - 1

, which simplifies to

6 - 5y = 3c - 1

, or

y = \frac{7 - 3c}{5}

. Substituting it into the second equation yields

(2 - y) + 3y = 2c - 8

, which simplifies to

2 + 2y = 2c - 8

, or

y = c - 5

. Equating these two expressions gives

c - 5 = \frac{7 - 3c}{5}

. Multiplying both sides by

5

results in

5c - 25 = 7 - 3c

. Adding

3c

and

25

to both sides yields

8c = 32

, which gives

c = 4

.

Adım Adım Çözüm

1

Express

x

in terms of

y

using the given constraint

x + y = 2

.

x = 2 - y

This allows us to substitute the relation into the system of equations and reduce the variables from two to one.

2

Substitute

x = 2 - y

into the first equation,

3x - 2y = 3c - 1

, and isolate

y

.

3(2 - y) - 2y = 3c - 1 \Rightarrow 6 - 5y = 3c - 1 \Rightarrow 5y = 7 - 3c \Rightarrow y = \frac{7 - 3c}{5}

To represent

y

as a function of the parameter

c

.

3

Substitute

x = 2 - y

into the second equation,

x + 3y = 2c - 8

, and isolate

y

.

(2 - y) + 3y = 2c - 8 \Rightarrow 2 + 2y = 2c - 8 \Rightarrow 2y = 2c - 10 \Rightarrow y = c - 5

To obtain another independent expression for

y

in terms of

c

.

4

Equate the two expressions for

y

and solve for

c

.

c - 5 = \frac{7 - 3c}{5} \Rightarrow 5(c - 5) = 7 - 3c \Rightarrow 5c - 25 = 7 - 3c \Rightarrow 8c = 32 \Rightarrow c = 4

Since both expressions represent the same value

y

, they must be equal, allowing us to determine the constant

c

.

Anahtar Kavram

Solving a system of linear equations containing unknown parameters under given linear constraints.

Alternatif Yöntem

Alternatively, substitute the relationship

x = 2 - y

into both equations to write them as a system in terms of

y

and

c

:

3(2 - y) - 2y = 3c - 1 \Rightarrow 5y + 3c = 7

and

(2 - y) + 3y = 2c - 8 \Rightarrow 2y - 2c = -10

. This simplified system can be solved for

c

by multiplying the first equation by 2 and the second equation by 5 to eliminate

y

:

2(5y + 3c) - 5(2y - 2c) = 2(7) - 5(-10) \Rightarrow 6c + 10c = 14 + 50 \Rightarrow 16c = 64 \Rightarrow c = 4

.

Tahmini Süre:2m 0s

Soru 1194Soru →

During the late Tang dynasty, elite scholars frequently gathered in private gardens to compose poetry, play music, and share their latest works of calligraphy. Although these gatherings were ostensibly informal, the participants used them to forge critical political alliances. Indeed, these ______ intellectual exchanges helped define the aesthetic standards of the era, ultimately elevating private artistic meetings into a major cultural institution that influenced generations of Chinese writers.

Which choice completes the text so that it conforms to the conventions of Standard English?

scholar

scholars

scholar's

scholars'

Cevabı ve açıklamayı göster

Cevap: scholars'

Cevap

scholars'

The correct answer is the plural possessive noun 'scholars\''. The plural demonstrative determiner 'these' requires a plural noun, ruling out the singular options. Furthermore, because this noun modifies the noun phrase 'intellectual exchanges' by showing who performed them, it must be in the possessive case. The correct plural possessive form is spelled with an apostrophe after the plural 's'.

Adım Adım Çözüm

1

Analyze the grammatical function of the blank within the sentence structure.

The word in the blank modifies the noun phrase 'intellectual exchanges,' which requires the possessive case to show association or ownership.

Nouns that function as modifiers for other noun phrases must be in the possessive case.

2

Determine the grammatical number of the noun based on nearby context clues.

The plural demonstrative determiner 'these' precedes the blank, requiring the noun it modifies to be plural.

Demonstrative determiners must agree in number with the nouns they modify.

3

Combine the plural and possessive requirements to identify the correct spelling and punctuation.

The plural possessive form of 'scholar' is 'scholars'', which places the apostrophe after the plural ending 's'.

Standard English conventions require regular plural possessive nouns to end with an apostrophe after the plural 's'.

Anahtar Kavram

Plural and Possessive Nouns

Soru 1195Soru →

For which values of $x$ is the inequality $2 - 3(x + 4) \geq 11$ true?

x \leq -7

x \geq -7

x \leq 1

x \leq -1

Cevabı ve açıklamayı göster

Cevap: $x \leq -7$

Cevap

The correct inequality is

x \leq -7

.

Subtracting 2 from both sides of

2 - 3(x + 4) \geq 11

gives

-3(x + 4) \geq 9

. Dividing both sides by the negative number

-3

requires reversing the inequality sign, yielding

x + 4 \leq -3

. Subtracting 4 from both sides completes the isolation of

x

, yielding the correct solution

x \leq -7

.

Adım Adım Çözüm

1

Subtract 2 from both sides of the inequality.

-3(x + 4) \geq 9

This isolates the term containing the parentheses on one side of the inequality.

2

Divide both sides of the inequality by

-3

and reverse the direction of the inequality sign.

x + 4 \leq -3

Dividing an inequality by a negative number requires reversing the direction of the inequality symbol to maintain equivalence.

3

Subtract 4 from both sides of the inequality.

x \leq -7

This isolates the variable

x

on one side of the inequality, providing the final solution set.

Anahtar Kavram

Solving multi-step linear inequalities in one variable, including distributing negative constants and reversing the inequality direction when dividing by a negative number.

Soru 1196Soru →

In the $xy$ -plane, the graph of a linear function $f$ has a $y$ -intercept of $(0, r)$ and an $x$ -intercept of $(s, 0)$ , where $r$ and $s$ are positive constants. The line $y = -2x$ is perpendicular to the line that passes through the origin $(0,0)$ and the midpoint of the segment connecting the two intercepts of $f$ . If $r = 12$ , what is the value of $s$ ?

Cevabı ve açıklamayı göster

Cevap: 24

Cevap

24

The midpoint of the segment connecting

(0, 12)

and

(s, 0)

is

(\frac{s}{2}, 6)

. The line passing through the origin

(0, 0)

and this midpoint has a slope of

\frac{6}{s/2} = \frac{12}{s}

. Since this line is perpendicular to the line

y = -2x

, which has a slope of

-2

, its slope must be the negative reciprocal of

-2

, which is

\frac{1}{2}

. Equating these two slopes, we get

\frac{12}{s} = \frac{1}{2}

, which simplifies to

s = 24

.

Adım Adım Çözüm

1

Find the coordinates of the intercepts and their midpoint.

The intercepts are

(0, 12)

and

(s, 0)

, and their midpoint is

(\frac{s}{2}, 6)

.

The intercepts of the function

f

form a line segment whose midpoint must be calculated.

2

Calculate the slope of the line passing through the origin and the midpoint.

The slope is

\frac{12}{s}

.

A line passing through the origin

(0,0)

and a point

(x_1, y_1)

has a slope of

\frac{y_1}{x_1}

.

3

Relate the slope of the line to the perpendicular line

y = -2x

.

The slope of the line must be

\frac{1}{2}

.

Perpendicular lines have slopes that are negative reciprocals of each other, and the negative reciprocal of

-2

is

\frac{1}{2}

.

4

Solve for the value of

s

.

s = 24

.

Setting the two expressions for the slope equal,

\frac{12}{s} = \frac{1}{2}

, yields

s = 24

.

Anahtar Kavram

Linear Functions and Graphs

Tahmini Süre:2m 30s

Soru 1197Soru →

A student wants to buy $x$ notebooks and $y$ pens. The student must buy at least $3$ notebooks. Each notebook costs $3$ dollars and each pen costs $2$ dollars. If the student can spend a maximum of $15$ dollars, what is the maximum number of pens the student can buy?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

3

The correct answer is 3. The constraints are represented by the system of inequalities

x \geq 3

and

3x + 2y \leq 15

. To find the maximum number of pens

y

, we minimize the number of notebooks

x

by setting

x = 3

. Substituting

x = 3

into the budget inequality gives

3(3) + 2y \leq 15

, which simplifies to

9 + 2y \leq 15

. Subtracting

9

from both sides results in

2y \leq 6

. Dividing by

2

yields

y \leq 3

. Therefore, the maximum number of pens the student can buy is 3.

Adım Adım Çözüm

1

Set up the system of inequalities from the given constraints.

The system of inequalities is

x \geq 3

and

3x + 2y \leq 15

, where

x

represents the number of notebooks and

y

represents the number of pens.

To represent the minimum number of notebooks required and the total budget limit.

2

Substitute the minimum possible value of

x

into the budget inequality to maximize

y

.

Setting

x = 3

gives

3(3) + 2y \leq 15

, which simplifies to

9 + 2y \leq 15

.

Since the cost per notebook is positive, minimizing the number of notebooks leaves the maximum amount of budget for purchasing pens.

3

Solve the inequality for

y

.

2y \leq 6 \implies y \leq 3

.

To find the upper bound for the number of pens the student can buy.

Anahtar Kavram

Solving a system of linear inequalities in a real-world context to find a maximum value.

Soru 1198Soru →

During the Middle Ages, master builders of Gothic cathedrals developed a structural innovation that allowed for unprecedented height and light _______ the flying buttress, which redirected the massive lateral forces of the stone vaults away from the walls and down to external supports. This engineering feat enabled the construction of soaring nave walls dominated by expansive stained-glass windows.

Which choice completes the text so that it conforms to the conventions of Standard English?

innovation, this was

innovation:

innovation;

innovation was

Cevabı ve açıklamayı göster

Cevap: innovation:

Cevap

The choice ending with a colon correctly introduces the explanatory appositive phrase that describes the structural innovation.

The correct choice properly uses a colon to link an independent clause ('During the Middle Ages... innovation that allowed for unprecedented height and light') to an explanatory appositive noun phrase ('the flying buttress...'). A colon must follow an independent clause and can introduce an appositive that clarifies or defines a term in the first clause.

Adım Adım Çözüm

1

Analyze the clause preceding the blank.

The clause 'During the Middle Ages, master builders of Gothic cathedrals developed a structural innovation that allowed for unprecedented height and light' is an independent clause.

Identifying if the preceding clause is independent is necessary because colons must be preceded by a grammatically complete sentence.

2

Analyze the phrase following the blank.

The phrase 'the flying buttress, which redirected the massive lateral forces of the stone vaults away from the walls and down to external supports' is a noun phrase functioning as an appositive.

Knowing that the subsequent phrase is a noun phrase (and not an independent clause) rules out the use of a semicolon.

3

Select the appropriate punctuation to link the independent clause to the appositive.

A colon is the correct punctuation mark to introduce an explanation, definition, or appositive after an independent clause.

Using a colon satisfies the syntactic constraints of the sentence while providing the correct logical link.

Anahtar Kavram

A colon must be preceded by an independent clause and can be used to introduce an explanation or appositive. A semicolon must separate two independent clauses.

Soru 1199Soru →

In the $xy$ -plane, a point $(x, y)$ is a solution to the system of inequalities below.

y \leq -3x + 14

y \leq 4x - 7

What is the maximum possible value of $y$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The maximum possible value of

y

is 5.

The maximum value of

y

is found at the intersection of the two boundary lines,

y = -3x + 14

and

y = 4x - 7

. Solving the system of equations gives

x = 3

and

y = 5

. Because the inequalities are both of the 'less than or equal to' type, any point in the solution set must lie on or below both lines. For

x < 3

, the value of

y

is limited by

y \leq 4x - 7

, which is strictly less than 5. For

x > 3

, the value of

y

is limited by

y \leq -3x + 14

, which is also strictly less than 5. Thus, the maximum value of

y

is 5.

Adım Adım Çözüm

1

Set the two boundary equations equal to find their intersection point.

-3x + 14 = 4x - 7

The maximum value of

y

under these constraints occurs at the intersection of the two boundary lines.

2

Solve the equation for

x

.

x = 3

Isolating

x

determines the x-coordinate of the intersection point.

3

Substitute

x = 3

back into one of the boundary equations to solve for

y

.

y = 5

This yields the y-coordinate of the intersection point, which is the upper bound for

y

in the solution set.

Anahtar Kavram

Determining the maximum value of a variable in the solution set of a system of linear inequalities by finding the intersection of their boundary lines.

Tahmini Süre:1m 30s

Soru 1200Soru →

In the $xy$ -plane, a line has an $x$ -intercept of $(k + 2, 0)$ and a $y$ -intercept of $(0, 2k + 2)$ , where $k$ is a positive constant. If the line passes through the point $(3, 5)$ , what is the value of $k$ ?

1.5

2.5

4

6.5

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

The correct answer is the value 4. By substituting the point

(3, 5)

into the intercept form of the line equation

\frac{x}{k+2} + \frac{y}{2k+2} = 1

, we obtain

\frac{3}{k+2} + \frac{5}{2k+2} = 1

. Multiplying by the common denominator

(k+2)(2k+2)

and simplifying results in the quadratic equation

2k^2 - 5k - 12 = 0

. Factoring this equation yields

(2k+3)(k-4) = 0

. Since

k

must be a positive constant, we select

k = 4

.

Adım Adım Çözüm

1

Write the equation of the line using intercept form.

The equation of a line with

x

-intercept

(k+2, 0)

and

y

-intercept

(0, 2k+2)

is

\frac{x}{k+2} + \frac{y}{2k+2} = 1

.

Intercept form

\frac{x}{a} + \frac{y}{b} = 1

is the most direct way to represent a line when both intercepts are given.

2

Substitute the given point

(3, 5)

into the line's equation.

Substituting

x = 3

and

y = 5

yields the equation

\frac{3}{k+2} + \frac{5}{2k+2} = 1

.

Since the line passes through

(3, 5)

, these coordinates must satisfy the equation of the line.

3

Eliminate the denominators by multiplying both sides of the equation by the least common denominator.

Multiplying by

(k+2)(2k+2)

gives

3(2k+2) + 5(k+2) = (k+2)(2k+2)

.

This clears the fractions and allows us to rewrite the equation in polynomial form.

4

Expand both sides of the equation and combine like terms.

Expanding both sides gives

6k + 6 + 5k + 10 = 2k^2 + 2k + 4k + 4 \implies 11k + 16 = 2k^2 + 6k + 4

.

To solve a polynomial equation, we need to simplify the algebraic expressions on both sides.

5

Rearrange the equation into standard quadratic form.

Moving all terms to one side gives

2k^2 - 5k - 12 = 0

.

Standard quadratic form

Ax^2 + Bx + C = 0

is required to apply factoring or the quadratic formula.

6

Factor and solve the quadratic equation.

(2k+3)(k-4) = 0

, which gives

k = -1.5

or

k = 4

.

Factoring determines the values of

k

that satisfy the quadratic relation.

7

Apply the constraint that

k

is a positive constant.

Since

k > 0

, we discard

k = -1.5

, leaving

k = 4

.

The problem specifies that

k

must be a positive constant.

Anahtar Kavram

Using the intercepts of a line to formulate its equation and solving the resulting rational/quadratic equation given a point on the line.

Alternatif Yöntem

Instead of using the intercept form of the line, you can equate the slope calculated between the y-intercept

(0, 2k+2)

and the point

(3,5)

to the slope calculated between the point

(3,5)

and the x-intercept

(k+2,0)

. This yields the equation

\frac{5 - (2k+2)}{3 - 0} = \frac{0 - 5}{(k+2) - 3}

, which simplifies to

\frac{3 - 2k}{3} = \frac{-5}{k - 1}

. Cross-multiplying and simplifying leads to the same quadratic equation,

2k^2 - 5k - 12 = 0

.

Tahmini Süre:2m 30s

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